Note on Logic Function and Boolean Algebra

  • Note
  • Things to remember

INTRODUCTION

Published in 1854, a book called “The Laws of Thought” described how humans thinksand make decisions. Complete with the practical mathematics of logic and probabilities, this book provided the rationale and methodology for reducing complex logical relationships to their simpler forms which can reproduce all the possible relationships from which the set was derived. The writer of the book was the famous English Mathematician, George Boole (1815-1864) and thus, this process is known as Boolean Reduction.

This process is used to reduce the size and complexity of complex digital logic designs to produce workable logic designs for digital devices’ circuits. In 1938, Claude Shannon, a mid-20thmathematician and philosopher, showed how the basic rules of logic, first given by George Boole in 1854 in his book, could be used to design digital circuits.

  • Boolean algebra is used to design and simplify circuits of electronic devices.
  • Each input and output are thought as a member of the set {0, 1}.
  • The basic elements of circuits are called gates. Each type of gate implements Boolean operation.


Boolean Algebra

Source: www.haikudeck.com
Source: www.haikudeck.com

Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations. It makes possible to transform logical statements into mathematical symbols and to calculate the truth or falsity of related statements by using rules. It is named after George Boole, a 19th-century Mathematician and Philosopher, who was the first to try and to formalize what we call logic or reasoning.

In computer science field, binary logic is referred as 'Digital Logic' which is considered as the heart of the operation of all modern digital computers. It shows the logical relationship between two or more logical functions. Instead of the use of T and F for true and false (which is frequently used for the truth tables) for the indication of the state of the sentences, Boolean algebra usually denotes it by 1 and 0 respectively. It is a branch of algebra in which the values of the variables are the truth values; true or false respectively.

This symbolic logic which was designed using variables and symbols is the language used to describe and manipulate logical statements and determine whether they are true or not. Through this, certain logical statements can be expressed in mathematical terms. For example, the symbols '•', '-' and '+' represent 'and', 'not'and 'or'respectively. Aside from its many applications in the design of computers, it serves as the foundation of information theory.


Boolean variable

A boolean variable is the variables which have only two states i.e. true/ false or right/ wrong or on/off or 0/1. As a computer is a binary system, it operates on an electronic signal which has only 2 possible states.

The signal that does not change its state with time is called constant signal and its value always remains the same i.e. either 1 or 0 whereas a variable signal continuously changes its state according to the time. At some point, the value of the variable signal may be 1 and at some another point, it might be 0.Therefore, these variables which consist of only two values i.e. 1 and 0 are Boolean variables or logic variables. These variables are denoted by English capital letters like A, B, X, Y, etc.


Boolean algebra as Switching algebra:

Boolean Algebra which is also known as the Switching algebra consists of two elements (0 and 1) has two binary operators called OR and AND and another unary operator called NOT. The unary operator performs on a single operand whereas a binary operator requires more than one operand.
Followings are their symbols:

Operator

Symbols

OR

~, +, V, U (Union)

AND

(.),^, Intersection

NOT

Ë…, (‘), (‾), Negation



Differences between Boolean algebra and Ordinary algebra:

Boolean algebra

Ordinary algebra

Its basic operations are AND, OR and NOT operations.

Its basic operations include addition, subtraction, multiplication, and division.

There is no exponents or coefficients involved in Boolean algebra i.e. A+A=A and A.A=A.

It consists of coefficients and exponents such as A+A=2A and A.A=A2.

It has only a finite set of elements. That is, it deals with only two elements; 0 and 1.

It deals with real numbers that contain an infinite number of elements (1, 2, 3…).

It holds both distributive laws: A.(B+C)=(A.B)+(A.C) and A+(B.C)=(A+B).(A+C)

It holds only one distributive law:
A.(B+C)=(A.B)+(A.C)

It is used in the field of digital electronics.

It is used in the field of mathematics.



Logic function (Boolean function)

Source: slideplayer.com
Source: slideplayer.com

Boolean function, commonly known as a logic function is an expression expressed algebraically with binary variables, logical operation symbols, parenthesis and equal sign. For a given value of the binary variables, the logic function can be either 0 or 1.

Example: Consider the logic function in algebraic expression:

F = X.Y.Z’+X.Y
Here, X, Y, Z are Boolean variables. The equation on the right-hand side above is known as an expression. Each occurrence of a variable or its complement in an expression is called literal.So, there are altogether three variables i.e. X, Y and Z and five literals; X, Y, Z’, X, and Y.

Logical circuit
By using transistors or Integrated Circuits (IC) or (LSI), the high and low-level voltage electrical signals called logical elements are produced. These logical elements are essential for the composition of a circuit for a specific operation called logical circuit.

Basic Logical/ Boolean Operation:

Introduction:

An operator is a special symbol that indicates the operation to be carried out between two operands. An operation is an action to be carried out upon operands. There are 3 basic Boolean Operations: AND, OR and NOT operations.

  • AND operation:
    Known as logical multiplication, it is carried out by dot (.) operator or simply by AND. If the inputs are true, it generates true output. Otherwise, it generates false output. Its logical equation is written as C=A.B or C=A AND B. The truth table of AND operation is:

Inputs

Output

A

B

C=A.B

False

False

False

False

True

False

False

False

False

True

True

False

  • OR operation
    Known as logical addition, it is carried out by plus (+) operator or simply by OR. If at least one input is true, it generates true output or else, it gives false output. The logical equation of OR operation is written as C=A+B or C=A OR B. The truth table of OR operation is given below:

Inputs

Output

A

B

C=A+B

False

False

False

False

True

True

True

False

True

True

True

True

  • NOT Operation
    Also known as the logical compliment, it is carried out by prime (‘) operator or bar (‾). It generates the output opposite the input i.e. if the input is true, it generates false output and vice versa. It's logical equation can be written as C=A’. The truth table of OR operation is:

Inputs

Output

A

C=A’

True

False

False

True

(Manandhar & Bhusal, 2013)


Bibliography

Manandhar, S., & Bhusal, R. (2013). Computer Essentials I. Putalisadak, Kathmandu: Asmita's Publication.

  • Boolean algebra is used to design and simplify circuits of electronic devices.
  • Each input and output are thought as a member of the set {0, 1}.
  • The basic elements of circuits are called gates. Each type of gate implements Boolean operation.
  • Boolean function or logic function is an expression expressed algebraically with binary variables,

 

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Very Short Questions

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Should lines 4 and 5 of "AND operation:" table read. A B C=A.B TRUE FALSE FALSE TRUE TRUE TRUEto account for all cases


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Why NAND gate and NOR gate are called universal gates?


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